OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 547 
Again, 
aS {(a— B)'(v—y)(w —8)(w—e)(w— Y} =86 | (0) , (9) 
(a) (fF) 
= 6’a’x the second of Sturm’s functions . ’ (22); 
the whole series of Sturm’s functions being F’(z), F,(x), F3(#),... . where 
6} (2) is the first derived function of F’(z), and the coefficient of the highest 
power of x in F(z) is unity. 
In like manner the second set of equations for a sextic gives rise to 
6 | (0) , 5(c) , LO(d) , 10(e) , 57) 5)» - |(28). 
1 |(a) ,5(b), 10) ,10(d), 5(e) ,(f) , » 
6]. .(@) 5) 10d), 10) ,5(f), ) 
6] .,(@ ,5() ,10(¢) ,10(d) ,5() , (f) 
a*{ C(aBy). (ON (Ger (O66 hig’ arly ne | — 
he Oe gilded: (eb )igiye 
1 , (©) , (8) , (def) 
Hence, for example, 
w'B\ 2aBy).(e—8) (e—2)(a—O} =6°| (0) , 5(c) , 10(d), (24), 
ay, o(O) > 10fe), 
» (4), 5(¢), (9) 
|. > 42), 50), (7) 
which may be written = 63 {A'(S)—A(g)}, 
where A’=5(c).A—(b).B and A= 5 {(b)?—(a)(o} 
A =50).A—(9.B B=10{6)() —(@(@)}, 
whence (a)A’= (bd) A— 5? {(b)? — (a) (0) }?.. 
Therefore we may write 
a®{....}=6| OM OD ga_sioyr—qoy.B 
Also (2) =a 
(b)’— (4) (¢)=0'—ae 
(0) (c)— (a) (d) = oe + (bc—ad) 
(b) (@)—(@ (@) =3(0? —ac)a’ + 8(bc—ad)x + (bd—ae) 
(0) (e) — ies 4(b° —ac)x’ + 6(be—ad)x’ + 4(bd —ae)a + (be—a/) , 
&e. 
A= JC0) nae 
10(bce—ad) ~ 
| QL+ 5b—a 5(0*—ac) \ ’ , (25), 
(OS) 
